Express their times in terms of the known distances and the variable
rates, as in Example 6.Cheryl travels 300 m at a rate of xmeters per second.
Cheryl’s timeBill travels 300 m at a rate of meters per second.
t= Bill’s time
d
r
=
300
x+ 1
1 x+ 12
t=
d
r
=
300
x
400 CHAPTER 7 Rational Expressions and Functions
Distance Rate TimeCheryl 300 xBill 300 300
x+ 1
x+ 1300
x
Bill’s time is 50 sec less
than Cheryl’s time.Step 3 Write an equation,using the times from the table.
Bill’s Cheryl’s less 50
time is time seconds.Step 4 Solve.
Distributive propertyMultiply.
Distributive property
Standard form
Divide by 50.
Factor.
Zero-factor property
x =- 3 or x= 2 Solve each equation.x+ 3 = 0 or x- 2 = 0
1 x+ 321 x- 22 = 0
x^2 + x- 6 = 0
50 x^2 + 50 x- 300 = 0
300 x= 300 x+ 300 - 50 x^2 - 50 x
300 x= 3001 x+ 12 - 50 x 1 x+ 12
x 1 x+ 12 a
300
x+ 1
b = x 1 x+ 12 a
300
x
b -x 1 x+ 121502
x 1 x+ 12 a
300
x+ 1
b = x 1 x+ 12 a
300
x
- 50 b
- 50
300
x
=
300
x+ 1
⎧⎨⎩ ⎧⎨⎩ ⎧⎪⎨⎪⎩
Multiply by the LCD,
x 1 x+ 12.Discard the negative answer, since rate (speed) cannot be negative.
Step 5 State the answer.Their walking rate is 2 m per sec.
Step 6 Checkthe solution in the words of the original problem. NOW TRY
CAUTION We cannot solve the rational equation in Example 7,
by using cross products. That method can be used only when there is a single ra-
tional expression on each side,such as in the equation in Example 6,
10
x- 3
=
15
x+ 3
.
300
x+ 1
=
300
x
- 50,
NOW TRY
EXERCISE 7
James and Pat are driving
from Atlanta to Jacksonville,
a distance of 310 mi. James,
whose average rate is 5 mph
faster than Pat’s, will drive the
first 130 mi of the trip and
then Pat will drive the rest of
the way to their destination. If
the total driving time is 5 hr,
determine the average rate of
each driver.
NOW TRY ANSWER
- 60 mph; 65 mph
To solve, multiply
by the LCD.To solve, multiply by the
LCD oruse cross products.